Problem 61 - Chemistry The Central Science In Si Units

Question

You make two solutions of a nonvolatile solute with a liquid solvent, \(0.01 M\) and \(1.00 M .\) Indicate whether each of the following statements is true or false. (a) The vapor pressure of the concentrated solution is higher than that of the diluted solution. (b) The osmotic pressure of the concentrated solution is higher than that of the diluted solution. (c) The boiling point of the concentrated solution is higher than that of the diluted solution. (d) The freezing point of the concentrated solution is higher than that of the diluted solution.

Step-by-Step Solution

Answer

Statements: (a) False, (b) True, (c) True, (d) False.

Step 1: Analyzing Vapor Pressure

Raoult's law states that the vapor pressure of a solution is lower than that of the pure solvent, and decreases with increasing solute concentration. Therefore, the vapor pressure of the 1.00 M solution is lower than that of the 0.01 M solution. Statement (a) is false.

Step 2: Evaluating Osmotic Pressure

Osmotic pressure is directly proportional to solute concentration according to the equation \( \Pi = iMRT \), where \( \Pi \) is the osmotic pressure, \( M \) is the molarity, \( R \) is the gas constant, and \( T \) is the temperature. Hence, the osmotic pressure of the 1.00 M solution is higher than that of the 0.01 M solution. Statement (b) is true.

Step 3: Analyzing Boiling Point Elevation

The boiling point elevation \( \Delta T_b \) is given by \( \Delta T_b = iK_bm \), where \( K_b \) is the ebullioscopic constant and \( m \) is the molality. Since boiling point elevation is proportional to solute concentration, the boiling point of the 1.00 M solution is higher than that of the 0.01 M solution. Statement (c) is true.

Step 4: Discussing Freezing Point Depression

Freezing point depression \( \Delta T_f \) is calculated with \( \Delta T_f = iK_fm \), where \( K_f \) is the cryoscopic constant. Freezing point depression also depends on concentration, with higher solute concentrations causing lower freezing points. Thus, the freezing point of the 1.00 M solution is lower than that of the 0.01 M solution. Statement (d) is false.

Key Concepts

Vapor Pressure Reduction

In simple terms, vapor pressure is the pressure exerted by a vapor in equilibrium with its liquid form. When a non-volatile solute is dissolved in a liquid solvent, the vapor pressure of the resulting solution is lower than that of the pure solvent. This phenomenon is predicted by Raoult's Law, which signifies that the presence of solute particles reduces the number of solvent molecules at the surface.
This means fewer molecules can escape into the vapor phase, thus lowering the vapor pressure. The more concentrated the solution, the greater the reduction in vapor pressure. Therefore, for our exercise, this means that a 1.00 M solution has a lower vapor pressure compared to a 0.01 M solution. This effect is crucial in understanding how solutes affect liquid behavior and is commonly observed in solutions containing non-volatile solutes.

Osmotic Pressure

Osmotic pressure is a colligative property that depends on the concentration of the solute particles, rather than their identity. It's essentially the pressure required to stop the flow of solvent molecules through a semi-permeable membrane from a dilute solution to a concentrated one.
The relationship is described by the formula:\[\Pi = iMRT\]where \( \Pi \) is the osmotic pressure, \( i \) is the van't Hoff factor, \( M \) is the molarity of the solution, \( R \) is the gas constant, and \( T \) is the temperature in Kelvin. This formula shows that osmotic pressure increases with solute concentration. In our example, a solution of 1.00 M has a higher osmotic pressure than a 0.01 M solution, as more solute particles in the higher concentration solution interact with the incoming solvent molecules, requiring more pressure to halt the process.

Boiling Point Elevation

Boiling point elevation occurs when the boiling point of a liquid is increased by the addition of a non-volatile solute. This is another colligative property that relies solely on the number of solute particles. It is explained by the formula:\[\Delta T_b = iK_bm\]In this formula, \( \Delta T_b \) is the boiling point elevation, \( i \) is the van't Hoff factor, \( K_b \) is the ebullioscopic constant unique to each solvent, and \( m \) is the molality of the solution. The presence of a solute interferes with the evaporation of solvent molecules, making it harder for the solution to reach its boiling point.
Thus, the boiling point rises. Applied to our situation, the 1.00 M solution boils at a higher temperature than the 0.01 M solution due to its higher solute concentration, reflecting more solute particles' impact on boiling dynamics.

Freezing Point Depression

Freezing point depression is the lowering of the freezing point of a liquid thanks to the presence of a solute. Like the other colligative properties, this effect is unaffected by the type of solute but directly connected to its concentration. The formula used is:\[\Delta T_f = iK_fm\]Here, \( \Delta T_f \) represents freezing point depression, \( i \) is the van't Hoff factor, \( K_f \) is the cryoscopic constant specific to the solvent, and \( m \) the solute's molality. When solute is added, it disrupts the structure necessary for a solid to form, effectively lowering the temperature at which the liquid will freeze.
In the exercise, the 1.00 M solution will have a lower freezing point than the 0.01 M solution. This illustrates that a higher concentration of solute results in greater depression, a principle fundamental to processes like de-icing roads using salt.